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A New Entropy Measure Based on Multiple Resolution and Noise Modeling Next: Data Retrieval Software for the USNO-A Catalog: Another Member of the -DARES Family
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Murtagh, F. & Starck, J.-L. 1999, in ASP Conf. Ser., Vol. 172, Astronomical Data Analysis Software and Systems VIII, eds. D. M. Mehringer, R. L. Plante, & D. A. Roberts (San Francisco: ASP), 403

A New Entropy Measure Based on Multiple Resolution and Noise Modeling

Fionn Murtagh
Queen's University Belfast and Strasbourg Observatory

Jean-Luc Starck
CEA, DAPNIA, Saclay

Abstract:

A new measure of information is defined which is based on noise modeling and incorporates resolution scale. Properties of this measure are discussed. Use of such an entropy measure for filtering and deconvolution is exemplified. Implications for structuring of, and access to, image archives are noted.

1. Introduction

We show how entropy can be incorporated into a multiscale image processing setting. The latter is a powerful setting for a wide range of image processing operations. Additionally it allows for image noise modeling, and subsequent noise removal.

Immediate applications include image filtering and deconvolution. Other objectives include object finding and definition, and feature characterization. More globally, we also note that such vision modeling will be necessary in future generation image databases.

Background on these methods can be found in Starck, Murtagh, & Bijaoui (1998). Information on a multiresolution image and vision software package, MR/1, can be found at http://visitweb.com/multires.

2. Multiscale Entropy Filtering Method

The multiscale entropy method consists of measuring the information h relative to wavelet coefficients, and of separating this into two parts hs, and hn. The expression hs is called the signal information and represents the part of h which is definitely not contaminated by the noise. The expression hn is called the noise information and represents the part of h which may be contaminated by the noise. We have h = hs+ hn. Following this notation, the corrected (i.e., filtered) wavelet or multiscale coefficient $\tilde w$ should minimize:

\begin{eqnarray*} J(\tilde w_j) = h_s(w_j-\tilde w_j) + \alpha h_n(\tilde w_j) \end{eqnarray*}


i.e. there is a minimum of information in the residual ($w-\tilde w$) due to the significant signal, and a minimum of information which can be due to the noise in the solution $ \tilde w_j$.

In order to verify a number of properties, the following functions are proposed for hs and hn in the case of Gaussian noise:

\begin{eqnarray*} h_s(w_j) & = & \frac{1}{\sigma_j^2} \int_{0}^{\mid w_j \mid} ... ...fc} \left( \frac{\mid w_j \mid -u}{\sqrt{2} \sigma_j} \right) du \end{eqnarray*}


3. Regularization Parameter and Data Model

The regularization parameter, $\alpha$, can be determined by using the fact that we expect a residual with a given standard deviation at each scale j equal to the noise standard deviation $\sigma_j$ at that scale. Hence we have an $\alpha_j$ per scale.

If we have a model, Dm, for the data we can use

\begin{eqnarray*} J_m(\tilde w_j) = h_s(w_j-\tilde w_j) + \alpha h_n(\tilde w_j - w_m) \end{eqnarray*}


4. Regularized Entropy-Based Filtering

The regularized entropy-based filtering algorithm is as follows.

  1. Estimate the noise in the data $\sigma$ (see Olsen 1993; Starck & Murtagh 1998a).
  2. Wavelet transform of the data.
  3. Calculate from $\sigma$ the noise standard deviation $\sigma_j$ at each scale j.
  4. Set $\alpha^{min}_j = 0$, $\alpha^{max}_j = 200$.
  5. For each scale j do
    1. Set $\alpha_j = \frac{\alpha^{min}_j + \alpha^{max}_j}{2}$
    2. For each wavelet coefficient wj,k of scale j, find $\tilde w_{j,k}$ by minimizing $j_m(\tilde w_{j,k})$ or $j_{ms}(\tilde w_{j,k})$
    3. Calculate the standard deviation of the residual:

      \begin{displaymath}\sigma_j^r = \sqrt{ \frac{1}{N_j} \sum_{k=1}^{N_j} (w_{j,k}-\tilde w_{j,k})^2}\end{displaymath}

    4. If $\sigma_j^r > \sigma_j$ then the regularization is too strong, and we set $\alpha^{max}_j$ to $\alpha_j$, otherwise we set $\alpha^{min}_j$ to $\alpha_j$.
  6. If $\alpha^{max}_j - \alpha^{min}_j > \epsilon $ then goto 5.
  7. Multiply all $\alpha_j$ by the constant $\alpha_u$ (default: 1).
  8. For each scale j and for each wavelet coefficient w find $\tilde w_{j,k}$ by minimizing $j_m(\tilde w_{j,k})$ or $j_{ms}(\tilde w_{j,k})$.
  9. Reconstruct the filtered image from $\tilde w_{j,k}$ by the inverse wavelet transform.

The minimization of jm or jms (step 5.2) can be done by any method. For instance, a simple dichotomy can be used in order to find $\tilde w$ such that

$\displaystyle \frac{\partial h_s(w-\tilde w)}{\partial \tilde w} = -\alpha_j \frac{\partial h_n(\tilde w)}{\partial \tilde w}$     (1)

The idea to treat the wavelet coefficients such that the residual respects some constraint has also been used in Nason (1996) and Amato & Vuza (1998) using cross-validation.

5. Filtering: Examples

Figure 1 shows a difficult case of smooth and sharp transitions. The filtering method described here allows an excellent noise filtering of it to be carried out. Figure 2 shows a spectrum, and an effective noise filtered version.

Figure 1: Top, noisy blocks and filtered blocks overplotted. Bottom, filtered blocks.
\begin{figure} \epsscale{0.4} \plotone{murtaghfd1.eps} \\ \plotone{murtaghfd2.eps} \end{figure}

Figure 2: Top, real spectrum and filtered spectrum overplotted. Bottom, filtered spectrum.
\begin{figure} \epsscale{0.4} \plotone{murtaghfd3.eps} \\ \plotone{murtaghfd4.eps} \end{figure}

6. Summary and Conclusion

We have described an integrated approach for specification of information content, handling of noise, and effective implementation. The multiscale entropy is similarly of benefit in deconvolution.

We note that large image repositories require

Hence information- and entropy-based processing technologies will be necessary for the effective design and implementation of large, future generation, image databases.

Further reading is available in Starck, Murtagh, & Gastaud (1998) and Starck & Murtagh (1998b). The methods described here are available in the multiresolution analysis software package, MR/1, Version 2.0. Details of the MR/1 software package can be found at http://visitweb.com/multires.

References

Amato, U. & Vuza, D. T. 1998, Rev. Roumaine Math. Pures Appl., in press

Nason, G. P. 1996, J. Roy. Stat. Soc. B, 58, 463

Olsen, S. I. 1993, Comp. Vis. Graph. Image Proc., 55, 319

Starck, J. L. & Murtagh, F. 1998a, PASP, 110, 193

\ibid , 1998b, Signal Proc., in press

Starck, J. L., Murtagh, F., & Bijaoui, A. 1998, Image and Data Analysis: The Multiscale Approach, (Cambridge: Cambridge Univ. Press)

Starck, J. L., Murtagh, F. & Gastaud, R. 1998, IEEE Trans. CAS II, 45, 1118

© Copyright 1999 Astronomical Society of the Pacific, 390 Ashton Avenue, San Francisco, California 94112, USA
Next: Data Retrieval Software for the USNO-A Catalog: Another Member of the -DARES Family
Up: Astrostatistics and Databases
Previous: Background Fluctuation Analysis from the Multiscale Entropy
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